Tubular approaches to Baker’s method for
curves and varieties

Fourn, Samuel Le

doi:10.2140/ant.2020.14.763

Cited by 5 publications

(3 citation statements)

References 17 publications

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“…The first explicit bounds for the heights of the solutions of (1.2) were established by Győry [11] using Baker's method. More recent work has partially replaced the use of Baker's method with Runge's method [15]. Building on this work, the current best bounds (in terms of S) for the heights of the solutions of (1.2) are due to [13].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Untitled

2021

Proc. Amer. Math. Soc. Ser. B

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Let K be a number field with ring of integers O K . We prove that if 3 does not divide [K : Q] and 3 splits completely in K, then there are no exceptional units in K. In other words, there are no x, y ∈ O × K with x + y = 1. Our elementary p-adic proof is inspired by the Skolem-Chabauty-Coleman method applied to the restriction of scalars of the projective line minus three points. Applying this result to a problem in arithmetic dynamics, we show that if f ∈ O K [x] has a finite cyclic orbit in O K of length n then n ∈ {1, 2, 4}.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Untitled

2021

Proc. Amer. Math. Soc. Ser. B

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“…In [19], Le Fourn uses a tubular method to give bounds for the heights of solutions of S ‐unit equations in terms of the norm of the third largest ideal in the set S . This bound cannot be directly used to prove Theorem 3 as the constants involved contain terms of the form

s^{s}

, where

s=| S |

.…”

Section: Application Of Győry's Recent Bound On the Solutions Of S‐un...mentioning

confidence: 99%

“…A combination of methods and results by Győry and Yu [13] and Győry [10, 11] with the method of Le Fourn [19] can be used directly to find results over the base field, as done by Győry in [12]. Further, in terms of S , Győry improved the S ‐unit bound given by Le Fourn.…”

Section: Some Remarksmentioning

confidence: 99%

On the abc$abc$ conjecture in algebraic number fields

Scoones

2023

Mathematika

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In this paper, we prove a weak form of the conjecture generalised to algebraic number fields. Given integers satisfying , Stewart and Yu were able to give an exponential bound in terms of the radical over the integers (Stewart and Yu [Math. Ann. 291 (1991), 225–230], Stewart and Yu [Duke Math. J. 108 (2001), no. 1, 169–181]), whereas Győry was able to give an exponential bound in the algebraic number field case for the projective height in terms of the radical for algebraic numbers (Győry [Acta Arith. 133 (2008), 281–295]). We generalise Stewart and Yu's method to give an improvement on Győry's bound for algebraic integers over the Hilbert Class Field of the initial number field K. Given algebraic integers in a number field K satisfying , we give an upper bound for the logarithm of the projective height in terms of norms of prime ideals dividing , where L is the Hilbert Class Field of K. In many cases, this allows us to give a bound in terms of the modified radical as given by Masser (Proc. Amer. Math. Soc. 130 (2002), no. 11, 3141–3150). Furthermore, by employing a recent bound of Győry (Publ. Math. Debrecen 94 (2019), 507–526) on the solutions of S‐unit equations, our estimates imply the upper bound where is an effectively computable constant. Further, given conditions on the largest prime ideal dividing , we obtain a sub‐exponential bound for in terms of the radical. Independently, as a direct application of his bounds on the solutions of S‐unit equations(Győry ([Publ. Math. Debrecen 94 (2019), 507–526]), Győry (Publ. Math. Debrecen 100 (2022), 499–511) also attains results mentioned above, including the above inequality, but over the base field K, as discussed in Section 6. As a consequence of our results, we will give an application to the effective Skolem–Mahler–Lech problem and give an improvement to a result by Lagarias and Soundararajan (J. Théor. Nombres Bordeaux 23 (2011), no. 1, 209–234) on the XYZ conjecture.

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Bounding integral points on the Siegel modular variety $$A_2(2)$$

Box

Fourn

2023

Res. number theory

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Tubular approaches to Baker’s method for curves and varieties

Cited by 5 publications

References 17 publications

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On the abc$abc$ conjecture in algebraic number fields

Bounding integral points on the Siegel modular variety $$A_2(2)$$

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